star wars - Why did Mace Windu threaten Jango Fett on Geonosis?

Inspired by this answer to the question Why did Jango Fett jump into the arena to engage Master Windu?

At the beginning of the Battle of Geonosis, Mace Windu had managed to sneak up on the Separatist leaders. He revealed his presence by activating his lightsaber and holding it to Jango Fett's throat. But why does he do this? I understand that it wouldn't have been the Jedi way for him to just start slicing them up from behind, but shouldn't Count Dooku have been his main target? What's more, Fett was a mercenary who was working for Dooku and the Separatists. Did Windu think that Dooku would care about his life?

Apparently, Mace Windu later regretted this decision. From Shatterpoint:

In my dreams, the purple flare of my blade sizzles the gray hairs of Dooku's beard, and in the critical semisecond it takes Jango Fett to aim and fire, I twitch that blade and take Dooku with me into death.

So why did he target Jango Fett instead of Count Dooku in the first place?

Answer

From what I can tell from the film's Official Novelisation, Mace Windu's decision to threaten Fett (rather than Dooku himself) seems to have boiled down to two reasons:

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  Although Dooku is a powerful ex-Jedi, Master Windu evidently judges Jango to be the deadlier enemy, noting his flamethrower and ability to fly, both of which are especially dangerous to a Jedi.

  
  
  


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  Gunray had just ordered Fett to kill Padmé. He wanted to ensure that that didn't happen.

  
  

“No!” Nute Gunray shouted back at him. “Jango, finish her off!”

Jango turned an amused expression Nute Gunray’s way, and nodded knowingly as Count Dooku motioned for him to stay put.

“Patience, Viceroy,” Dooku said to the fuming Gunray. “She will die.”

Even as he spoke, even as Gunray seemed about to explode with rage, the Count motioned back to the arena, and the Neimoidian turned to see a group of droidekas roll out from the side paddock. They surrounded the reek and the three prisoners and opened and unfolded into their battle position, giving Anakin no choice but to pull back hard on the makeshift rein and halt the creature.

“You see?” Dooku calmly asked.

The Count’s expression changed, though, just for a moment, as a familiar hum began right behind him. He glanced to his right quickly, to see a purple lightsaber blade right beside Jango Fett’s neck, then turned slowly to regard the wielder.

The film's junior novelisation also makes it clear that Fett is present as Dooku's bodyguard. Immobilising him would seem a sensible first step to capturing his master.

The Viceroy snorted and turned back to the arena. Amusing as it was to watch his frustration, it was time to end things. Dooku signaled surreptitiously, and his hidden droidekas poured from gates all around the arena. The crowd cheered and Nute Gunray sat back in satisfaction, but Dooku sensed only a feeling of chagrin from behind them where Jango Fett was standing. He turned to see what his bodyguard had noticed.